ishil2

Description: The predicate "is a Hilbert space" (over a *-division ring). (Contributed by NM, 7-Oct-2011) (Revised by Mario Carneiro, 22-Jun-2014)

	Ref	Expression
Hypotheses	ishil2.v	⊢ 𝑉 = ( Base ‘ 𝐻 )
	ishil2.s	⊢ ⊕ = ( LSSum ‘ 𝐻 )
	ishil2.o	⊢ ⊥ = ( ocv ‘ 𝐻 )
	ishil2.c	⊢ 𝐶 = ( ClSubSp ‘ 𝐻 )
Assertion	ishil2	⊢ ( 𝐻 ∈ Hil ↔ ( 𝐻 ∈ PreHil ∧ ∀ 𝑠 ∈ 𝐶 ( 𝑠 ⊕ ( ⊥ ‘ 𝑠 ) ) = 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		ishil2.v	⊢ 𝑉 = ( Base ‘ 𝐻 )
2		ishil2.s	⊢ ⊕ = ( LSSum ‘ 𝐻 )
3		ishil2.o	⊢ ⊥ = ( ocv ‘ 𝐻 )
4		ishil2.c	⊢ 𝐶 = ( ClSubSp ‘ 𝐻 )
5		eqid	⊢ ( proj ‘ 𝐻 ) = ( proj ‘ 𝐻 )
6	5 4	ishil	⊢ ( 𝐻 ∈ Hil ↔ ( 𝐻 ∈ PreHil ∧ dom ( proj ‘ 𝐻 ) = 𝐶 ) )
7	5 4	pjcss	⊢ ( 𝐻 ∈ PreHil → dom ( proj ‘ 𝐻 ) ⊆ 𝐶 )
8		eqss	⊢ ( dom ( proj ‘ 𝐻 ) = 𝐶 ↔ ( dom ( proj ‘ 𝐻 ) ⊆ 𝐶 ∧ 𝐶 ⊆ dom ( proj ‘ 𝐻 ) ) )
9	8	baib	⊢ ( dom ( proj ‘ 𝐻 ) ⊆ 𝐶 → ( dom ( proj ‘ 𝐻 ) = 𝐶 ↔ 𝐶 ⊆ dom ( proj ‘ 𝐻 ) ) )
10	7 9	syl	⊢ ( 𝐻 ∈ PreHil → ( dom ( proj ‘ 𝐻 ) = 𝐶 ↔ 𝐶 ⊆ dom ( proj ‘ 𝐻 ) ) )
11		dfss3	⊢ ( 𝐶 ⊆ dom ( proj ‘ 𝐻 ) ↔ ∀ 𝑠 ∈ 𝐶 𝑠 ∈ dom ( proj ‘ 𝐻 ) )
12	10 11	bitrdi	⊢ ( 𝐻 ∈ PreHil → ( dom ( proj ‘ 𝐻 ) = 𝐶 ↔ ∀ 𝑠 ∈ 𝐶 𝑠 ∈ dom ( proj ‘ 𝐻 ) ) )
13		eqid	⊢ ( LSubSp ‘ 𝐻 ) = ( LSubSp ‘ 𝐻 )
14	4 13	csslss	⊢ ( ( 𝐻 ∈ PreHil ∧ 𝑠 ∈ 𝐶 ) → 𝑠 ∈ ( LSubSp ‘ 𝐻 ) )
15	1 13 3 2 5	pjdm2	⊢ ( 𝐻 ∈ PreHil → ( 𝑠 ∈ dom ( proj ‘ 𝐻 ) ↔ ( 𝑠 ∈ ( LSubSp ‘ 𝐻 ) ∧ ( 𝑠 ⊕ ( ⊥ ‘ 𝑠 ) ) = 𝑉 ) ) )
16	15	baibd	⊢ ( ( 𝐻 ∈ PreHil ∧ 𝑠 ∈ ( LSubSp ‘ 𝐻 ) ) → ( 𝑠 ∈ dom ( proj ‘ 𝐻 ) ↔ ( 𝑠 ⊕ ( ⊥ ‘ 𝑠 ) ) = 𝑉 ) )
17	14 16	syldan	⊢ ( ( 𝐻 ∈ PreHil ∧ 𝑠 ∈ 𝐶 ) → ( 𝑠 ∈ dom ( proj ‘ 𝐻 ) ↔ ( 𝑠 ⊕ ( ⊥ ‘ 𝑠 ) ) = 𝑉 ) )
18	17	ralbidva	⊢ ( 𝐻 ∈ PreHil → ( ∀ 𝑠 ∈ 𝐶 𝑠 ∈ dom ( proj ‘ 𝐻 ) ↔ ∀ 𝑠 ∈ 𝐶 ( 𝑠 ⊕ ( ⊥ ‘ 𝑠 ) ) = 𝑉 ) )
19	12 18	bitrd	⊢ ( 𝐻 ∈ PreHil → ( dom ( proj ‘ 𝐻 ) = 𝐶 ↔ ∀ 𝑠 ∈ 𝐶 ( 𝑠 ⊕ ( ⊥ ‘ 𝑠 ) ) = 𝑉 ) )
20	19	pm5.32i	⊢ ( ( 𝐻 ∈ PreHil ∧ dom ( proj ‘ 𝐻 ) = 𝐶 ) ↔ ( 𝐻 ∈ PreHil ∧ ∀ 𝑠 ∈ 𝐶 ( 𝑠 ⊕ ( ⊥ ‘ 𝑠 ) ) = 𝑉 ) )
21	6 20	bitri	⊢ ( 𝐻 ∈ Hil ↔ ( 𝐻 ∈ PreHil ∧ ∀ 𝑠 ∈ 𝐶 ( 𝑠 ⊕ ( ⊥ ‘ 𝑠 ) ) = 𝑉 ) )

Metamath Proof Explorer

Theorem ishil2

Proof